To show that \( \triangle ABC \cong \triangle EFD \) using rigid transformations, we can follow a sequence of transformations that demonstrates the congruence of the triangles.
Steps for Rigid Transformations
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Translation: First, we can translate \( \triangle ABC \) so that point \( A \) coincides with point \( E \). During this translation, all points in \( \triangle ABC \) move the same distance in the same direction, hence preserving the lengths of all sides and the angles of the triangle.
- After this translation, point \( A \) overlaps with point \( E \).
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Rotation: Next, we can rotate \( \triangle ABC \) about point \( E \) so that side \( AB \) aligns with side \( EF \). We need to ensure that the angle formed at \( A \) (which is \( \angle A \)) matches with the angle formed at \( E \) (which is \( \angle E \)) upon rotation.
- After this rotation, if done correctly, point \( B \) will be at the position of point \( F \).
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Reflection (if necessary): If \( \triangle ABC \) is oriented differently from \( \triangle EFD \), we may need to perform a reflection across a line that goes through point \( E \) and is perpendicular to the line \( EF \) to ensure that the orientation matches.
- After the reflection (if applied), point \( C \) will now overlap with point \( D \).
Corresponding Vertices
After performing these rigid transformations, we will have:
- \( A \) overlaps with \( E \)
- \( B \) overlaps with \( F \)
- \( C \) overlaps with \( D \)
Since we have moved the triangles using rigid transformations, the corresponding sides and angles will perfectly match, confirming that \( \triangle ABC \) and \( \triangle EFD \) are indeed congruent.
Explanation of SAS Congruence Theorem
The sequence of transformations illustrates the SAS (Side-Angle-Side) Congruence Theorem. This theorem states that if two triangles have two sides that are equal in length and the angle between those sides is equal, then the triangles are congruent.
- In our scenario, by translating triangle \( ABC \) so that \( A \) coincides with \( E \), the lengths \( AB \) and \( EF \) correspond, and we have \( \angle A \) corresponding to \( \angle E \).
- We also ensured that the third sides \( AC \) and \( ED \) remain equal due to the properties of rigid transformations.
Thus, we have demonstrated that \( \triangle ABC \) and \( \triangle EFD \) are congruent by verifying that two sides and the included angle are congruent between the two triangles, fulfilling the conditions of the SAS Theorem.